The Stacks project

Remark 10.69.7 (Koszul regular sequences). In the paper [Kabele] the author introduces two more regularity conditions for sequences $x_1, \ldots , x_ r$ of elements of a ring $R$. Namely, we say the sequence is Koszul-regular if $H_ i(K_{\bullet }(R, x_{\bullet })) = 0$ for $i \geq 1$ where $K_{\bullet }(R, x_{\bullet })$ is the Koszul complex. The sequence is called $H_1$-regular if $H_1(K_{\bullet }(R, x_{\bullet })) = 0$. If $R$ is a local ring (possibly non-Noetherian) and the sequence consists of elements of the maximal ideal, then one has the implications regular $\Rightarrow $ Koszul-regular $\Rightarrow $ $H_1$-regular $\Rightarrow $ quasi-regular. By examples the author shows that these implications cannot be reversed in general. We introduce these notions in more detail in More on Algebra, Section 15.30.

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 061T. Beware of the difference between the letter 'O' and the digit '0'.